Confidence intervals for projections of partially identified parameters

We propose a bootstrap-based calibrated projection procedure to build con fidence intervals for single components and for smooth functions of a partially identi fied parameter vector in moment (in)equality models. The method controls asymptotic coverage uniformly over a large class of data generating processes. The extreme points of the calibrated projection confi dence interval are obtained by extremizing the value of the component (or function) of interest subject to a proper relaxation of studentized sample analogs of the moment (in)equality conditions. The degree of relaxation, or critical level, is calibrated so that the component (or function) of , not itself, is uniformly asymptotically covered with prespeci ed probability. This calibration is based on repeatedly checking feasibility of linear programming problems, rendering it computationally attractive. Nonetheless, the program defi ning an extreme point of the confi dence interval is generally nonlinear and potentially intricate. We provide an algorithm, based on the response surface method for global optimization, that approximates the solution rapidly and accurately. The algorithm is of independent interest for inference on optimal values of stochastic nonlinear programs. We establish its convergence under conditions satisfi ed by canonical examples in the moment (in)equalities literature. Our assumptions and those used in the leading alternative approach (a profi ling based method) are not nested. An extensive Monte Carlo analysis con rms the accuracy of the solution algorithm and the good statistical as well as computational performance of calibrated projection, including in comparison to other methods.